In a roundabout way, a jump mare is the female counterpart to a teaser - both are there to make sure the opposite sex is able to do their job. Teasers make sure a mare is ready TO breed and (in the case of an inexperienced stud) jump mares make sure a stallion knows HOW to breed .
Farms use jump mares/test mares to help one learn to breed and to test if they are able to impregnate a mare. Some of the mares have had their ovaries removed so they cannot get pregnant, and some mares they just abort should they get pregnant. It all depends on the stallion/farm manager’s particular needs. I owned a mare who was a real whore and she loved to teach the boys their job. She was 17 hands so on occasion we ran into a problem with a smaller stud but given time, they always figure it out 
The biggest thing to remember when training a breeding stallion is to be patient. For most of them, during their training/racing years, have been smacked or heavily reprimanded any time they showed any sort of interest in a filly and may be apprehensive to show any interest now when they smell a mare. Some learn their new job faster than others. Some want to learn on their own. A farm I knew that had a new stallion was having problems getting him to want to jump the mare and had spent several hours over the course of 2 days working with him. I suggested just turning the stud and mare out in a small paddock together for the afternoon to see if that would help. It worked like a charm - he knew his job from that day forward, and they never had a problem getting him to jump a mare since!!!
You asked about nurse mares and their well bred babies. I used to work in the office on one of the large farms in Lexington and one year we had the hands down, most beautiful palomino for a nurse mare. They bred her after hours to our best looking (and most expensive) stallion and the manager made up some story about how he liked that mare so much, and that he was looking for a riding horse to train and could he purchase her foal once it was weaned. Talk about a good looking baby!